scholarly journals Shallow‐water equations with complete Coriolis force: Group properties and similarity solutions

2021 ◽  
Author(s):  
Andronikos Paliathanasis
Keyword(s):  
Shallow Water ◽  
Coriolis Force ◽  
Shallow Water Equations ◽  
Similarity Solutions ◽  
Complete Coriolis Force

Multilayer shallow water equations with complete Coriolis force. Part 3. Hyperbolicity and stability under shear

2013 ◽  
Vol 723 ◽  
pp. 289-317 ◽  
Author(s):  
Andrew L. Stewart ◽  
Paul J. Dellar
Keyword(s):  
Shallow Water ◽  
Coriolis Force ◽  
Shallow Water Equations ◽  
Three Dimensional ◽  
Direct Calculation ◽  
Shear Instability ◽  
Vertical Shear ◽  
Vortex Sheets ◽  
Multilayer Shallow Water Equations ◽  
Complete Coriolis Force

AbstractWe analyse the hyperbolicity of our multilayer shallow water equations that include the complete Coriolis force due to the Earth’s rotation. Shallow water theory represents flows in which the vertical shear is concentrated into vortex sheets between layers of uniform velocity. Such configurations are subject to Kelvin–Helmholtz instabilities, with arbitrarily large growth rates for sufficiently short-wavelength disturbances. These instabilities manifest themselves through a loss of hyperbolicity in the shallow water equations, rendering them ill-posed for the solution of initial value problems. We show that, in the limit of vanishingly small density difference between the two layers, our two-layer shallow water equations remain hyperbolic when the velocity difference remains below the same threshold that also ensures the hyperbolicity of the standard shallow water equations. Direct calculation of the domain of hyperbolicity becomes much less tractable for three or more layers, so we demonstrate numerically that the threshold for the velocity differences, below which the three-layer equations remain hyperbolic, is also unchanged by the inclusion of the complete Coriolis force. In all cases, the shape of the domain of hyperbolicity, which extends outside the threshold, changes considerably. The standard shallow water equations only lose hyperbolicity due to shear parallel to the direction of wave propagation, but the complete Coriolis force introduces another mechanism for loss of hyperbolicity due to shear in the perpendicular direction. We demonstrate that this additional mechanism corresponds to the onset of a transverse shear instability driven by the non-traditional components of the Coriolis force in a three-dimensional continuously stratified fluid.

Consistent shallow-water equations on the rotating sphere with complete Coriolis force and topography

2014 ◽  
Vol 748 ◽  
pp. 789-821 ◽  
Author(s):  
Marine Tort ◽  
Thomas Dubos ◽  
François Bouchut ◽  
Vladimir Zeitlin
Keyword(s):  
Shallow Water ◽  
Coriolis Force ◽  
Shallow Water Equations ◽  
Primitive Equations ◽  
Conservation Of Energy ◽  
Rotating Sphere ◽  
Planetary Rotation ◽  
Dependent Change ◽  
Body Velocity ◽  
Complete Coriolis Force

AbstractConsistent shallow-water equations are derived on the rotating sphere with topography retaining the Coriolis force due to the horizontal component of the planetary angular velocity. Unlike the traditional approximation, this ‘non-traditional’ approximation captures the increase with height of the solid-body velocity due to planetary rotation. The conservation of energy, angular momentum and potential vorticity are ensured in the system. The caveats in extending the standard shallow-water wisdom to the case of the rotating sphere are exposed. Different derivations of the model are possible, being based, respectively, on (i) Hamilton’s principle for primitive equations with a complete Coriolis force, under the hypothesis of columnar motion, (ii) straightforward vertical averaging of the ‘non-traditional’ primitive equations, and (iii) a time-dependent change of independent variables in the primitive equations written in the curl (‘vector-invariant’) form, with subsequent application of the columnar motion hypothesis. An intrinsic, coordinate-independent form of the non-traditional equations on the sphere is then given, and used to derive hyperbolicity criteria and Rankine–Hugoniot conditions for weak solutions. The relevance of the model for the Earth’s atmosphere and oceans, as well as other planets, is discussed.

Multilayer shallow water equations with complete Coriolis force. Part 2. Linear plane waves

2011 ◽  
Vol 690 ◽  
pp. 16-50 ◽  
Author(s):  
Andrew L. Stewart ◽  
Paul J. Dellar
Keyword(s):  
Shallow Water ◽  
Internal Waves ◽  
Coriolis Force ◽  
Shallow Water Equations ◽  
Plane Waves ◽  
Vertical Component ◽  
Momentum Equation ◽  
Layer System ◽  
Multilayer Shallow Water Equations ◽  
Complete Coriolis Force

AbstractWe investigate the behaviour of linear plane waves in multilayer shallow water equations that include a complete treatment of the Coriolis force. These equations improve upon the conventional shallow water equations, based on the traditional approximation, that include only the part of the Coriolis force due to the locally vertical component of the rotation vector. Including the complete Coriolis force leads to dramatic changes in the structure of long linear plane waves. It allows subinertial waves to exist with frequencies below the inertial frequency, the minimum frequency for which waves exist under the traditional approximation. These subinertial waves are characterized by a distinguished limit in which the horizontal pressure gradient becomes comparable to the upwellings and downwellings driven by the non-traditional Coriolis term in the vertical momentum equation. The subinertial waves connect wave modes that remain separate in the conventional multilayer shallow water equations, such as the surface and internal waves in a two-layer system. Eastward-propagating surface waves in a two-layer system connect with westward-propagating internal waves, and vice versa, via the long subinertial waves. The long subinertial waves cannot be classified as either surface or internal waves, due to the phase difference between the disturbances to the interfaces in these waves.

scholarly journals Numerical approximation of the shallow water equations with coriolis source term

2021 ◽  
Vol 70 ◽  
pp. 31-44
Author(s):  
E. Audusse ◽  
V. Dubos ◽  
A. Duran ◽  
N. Gaveau ◽  
Y. Nasseri ◽  
...  
Keyword(s):  
Shallow Water ◽  
Finite Volume ◽  
Coriolis Force ◽  
Shallow Water Equations ◽  
Numerical Approximation ◽  
Energy Estimates ◽  
Discrete Energy ◽  
Geostrophic Balance ◽  
Low Froude Number ◽  
Numerical Fluxes

We investigate in this work a class of numerical schemes dedicated to the non-linear Shallow Water equations with topography and Coriolis force. The proposed algorithms rely on Finite Volume approximations formulated on collocated and staggered meshes, involving appropriate diffusion terms in the numerical fluxes, expressed as discrete versions of the linear geostrophic balance. It follows that, contrary to standard Finite-Volume approaches, the linear versions of the proposed schemes provide a relevant approximation of the geostrophic equilibrium. We also show that the resulting methods ensure semi-discrete energy estimates. Numerical experiments exhibit the efficiency of the approach in the presence of Coriolis force close to the geostrophic balance, especially at low Froude number regimes.

Modelling the Azov Sea circulation and extreme surges in 2013-2014 using the regularized shallow water equations

2018 ◽  
Vol 33 (3) ◽  
pp. 173-185 ◽  
Author(s):  
Dmitry S. Saburin ◽  
Tatiana G. Elizarova
Keyword(s):  
Shallow Water ◽  
Numerical Method ◽  
Coriolis Force ◽  
Shallow Water Equations ◽  
Numerical Algorithms ◽  
Water Area ◽  
Observation Data ◽  
Wind Effects ◽  
New Model ◽  
Azov Sea

Abstract A new model for calculation of circulation in shallow water basins is created based on the shallow water equations taking into account the Coriolis force and quadratic friction on the bottom. Wind effects are taken into account as forcing. The main feature of the model is a new numerical method based on regularized shallow water equations allowing one to construct the simple and sufficiently accurate numerical algorithms possessing a number of advantages over existing methods. The paper provides a detailed description of all construction steps of the model. The developed model was implemented for the water area of the Azov Sea. The paper presents the modelling of extreme surges in March 2013 and September 2014, the results of calculations are compared with observation data of hydrometeorological stations in Taganrog and Yeysk.

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